DaeKil Park
The non-relativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in $D$-dimensional free particle and harmonic oscillator systems. The Feynman propagators for these systems are exactly derived within the first order of the GUP parameter.
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We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354]. In particular, we show that the metric operators computed for the deformed non-Hermitian Swanson models differs in both cases and is inconsistent in the former.
Dependency on the preparation of state for the Heisenberg uncertainty principle can be removed with the help of entropy uncertainty principle. The shortness of the uncertainty principle (UP) can be overcome with the help of the concept of Shannon's information entropy (SE). In this article, we have shown that UP in terms of SE holds for a position-dependent effective mass system. We have considered the Dirac system with a mass-jump at the origin. We have proved the existence of a lower bound for a UP for this position-dependent effective mass.
A more general measurement disturbance uncertainty principle is presented in a Robertson-Schrödinger formulation. It is shown that it is stronger and having nicer properties than Ozawa's uncertainty relations. In particular is invariant under symplectic transformations. One shows also that there are states of the probe (measuring device) that saturate the matrix formulation of measurement disturbance uncertainty principle.
In this work we derive a matrix formulation of a noise-disturbance uncertainty relation, which is akin to the Robertson-Schrödinger uncertainty principle. Our inequality is stronger than Ozawa's uncertainty principle and takes noise-disturbance correlations into account. Moreover, we show that, for certain types of measurement interactions, it is covariant with respect to linear symplectic transformations of the noise and disturbance operators.
Within the formulation of a q-deformed Quantum Mechanics a qualitative undercut of the q-deformed uncertainty relation from the Heisenberg uncertainty relation is revealed. When $q$ is some fixed value not equal to one, recovering of ordinary quantum mechanics and the corresponding recovering condition are discussed.
We explore the modification of the entropic formulation of uncertainty principle in quantum mechanics which measures the incompatibility of measurements in terms of Shannon entropy. The deformation in question is the type so called generalized uncertainty principle that is motivated by thought experiments in quantum gravity and string theory and is characterized by a parameter of Planck scale. The corrections are evaluated for small deformation parameters by use of the Gaussian wave function and numerical calculation. As the generalized uncertainty principle has proven to be useful in the stud…
In this letter, we provide evidence for a classical sector of states in the Hilbert space of Finite Quantum Mechanics (FQM). We construct a subset of states whose the minimum bound of position -momentum uncertainty (equivalent to an effective $\hbar$) vanishes. The classical regime, contrary to standard Quantum Mechanical Systems of particles and fields, but also of strings and branes appears in short distances of the order of the lattice spacing. {}For linear quantum maps of long periods, we observe that time evolution leads to fast decorrelation of the wave packets, phenomenon similar to the…
In the present article, we discuss one of the basic relations of Quantum Mechanics - the Uncertainty Relation (UR). In 1930, few years after Heisenberg, Erwin Schrodinger generalized the famous Uncertainty Relation in Quantum Mechanics, making it more precise than the original. The present study discusses recent generalizations of Schrodinger's work and explains why his paper remains almost forgotten in the last century.
It is shown that the results of ref [1] are consistent.
The uncertainty relations for the position and momentum of a quantum particle on a circle are identified minimized by the corresponding coherent states. The sqeezed states in the case of the circular motion are introduced and discussed in the context of the uncertainty relations.
Uncertainty relations are shown to have nothing specific for quantum mechanics, being the general property valid for arbitrary function. A wave function of a particle having precisely defined position and momentum in QM simultaneously is demonstrated. Interference on two slits in a screen is shown to exist in classical mechanics. A nonlinear classical system of equations replacing QM Schrödinger equation is suggested. This approach is shown to have nothing in common with Bohmian mechanics.
The generalized uncertainty principle within the ordinary framework of quantum mechanics
针对广义不确定性原理对应的形变对易关系会脱离量子力学正则形式的问题，研究证明位置与动量算符的形变代数可纳入普通量子力学框架，为小尺度引力影响的唯象研究提供形式兼容的路径。
We study non-Hermitian quantum mechanics in the presence of a minimal length. In particular we obtain exact solutions of a non-Hermitian displaced harmonic oscillator and the Swanson model with minimal length uncertainty. The spectrum in both the cases are found to be real. It is also shown that the models are $η$ pseudo-Hermitian and the metric operator is found explicitly in both the cases.
A more detailed derivation of the Heisenberg uncertainty principle from the certainty principle is given.
Two central concepts of quantum mechanics are Heisenberg's uncertainty principle, and a subtle form of non-locality that Einstein famously called ``spooky action at a distance''. These two fundamental features have thus far been distinct concepts. Here we show that they are inextricably and quantitatively linked. Quantum mechanics cannot be more non-local with measurements that respect the uncertainty principle. In fact, the link between uncertainty and non-locality holds for all physical theories.More specifically, the degree of non-locality of any theory is determined by two factors -- the s…
Refinement of Uncertainty Relations in Quantum Mechanics
本文利用对易子和Pauli矩阵研究量子力学中三阶对易子贡献，改进了Heisenberg不确定性关系，并将其应用于一维系统动能、坐标及角动量分量。
We present a scheme in which we investigate the two-slit experiment and we show that the principle of complementarity is more fundamental then the uncertainty principle.
Mechanics can be founded on a principle relating the uncertainty delta-q in the trajectory of an observable particle to its motion relative to the observer. From this principle, p.delta-q=const., p being the q-conjugated momentum, mechanical laws are derived and the meaning of the Lagrangian and Hamiltonian functions are discussed. The connection between the presented principle and Hamilton's Least Action Principle is examined. Wave mechanics and Schrodinger equation appear without additional assumptions by choosing the representation for delta-q in the case the motion is not trajectory descri…
The emergence of the generalized uncertainty principle and the existence of a non-zero minimal length are intertwined. On the other hand, the Heisenberg uncertainty principle forms the core of the EPR paradox. Subsequently, here, the implications of resorting to the generalized uncertainty principle (or equally, the minimal length) instead of the Heisenberg uncertainty principle on the quantum non-locality are investigated through focusing on the Franson experiment in which the uncertainty relation is the backbone of understanding and explaining the results.
It is shown that polymer quantization leads to a modified uncertainty principle similar to that obtained from string theory and non-commutative geometry. When applied to quantum field theory on general background spacetimes, corrections to the uncertainty principle acquire a metric dependence. For Friedmann-Robertson-Walker cosmology this translates to a scale factor dependence which gives a large effect in the early universe.
In the preceding Comment (quant-ph/0209032) Trifonov disputes our recently proposed uncertainty relations for a quantum particle on a circle. He states that (i) the quantity $Δ^2(\hatϕ)$ introduced by us representing the uncertainty of the angle is not a proper measure of the position uncertainty and therefore the proposed inequalities can hardly be qualified as a relevant uncertainty relations on a circle; and that (ii) the most suitable uncertainty relations on a circle are those based on the Gram-Robertson matrix. It is shown by examples that both points are erroneous.
Heisenberg showed in the early days of quantum theory that the uncertainty principle follows as a direct consequence of the quantization of electromagnetic radiation in the form of photons. As we show here the gravitational interaction of the photon and the particle being observed modifies the uncertainty principle with an additional term. From the modified or gravitational uncertainty principle it follows that there is an absolute minimum uncertainty in the position of any particle, of order of the Planck length. A modified uncertainty relation of this form is a standard result of superstring…
We propose the experimental test of the uncertainty principle. From sub-quantum models it follows that the uncertainty principle may be not true on short time intervals of the order of a picosecond. The positive result of this experiment would signify the limits of QM.
In the Copenhagen interpretation the Heisenberg uncertainty relation is interpreted as the mathematical expression of the concept of complementarity, quantifying the mutual disturbance necessarily taking place in a simultaneous or joint measurement of incompatible observables. This interpretation has already been criticized by Ballentine a long time ago, and has recently been challenged in an experimental way. These criticisms can be substantiated by using the generalized formalism of positive operator-valued measures, from which a new inequality can be derived, precisely illustrating the Cope…
The Generalized Uncertainty Principle (GUP) has been directly applied to the motion of (macroscopic) test bodies on a given space-time in order to compute corrections to the classical orbits predicted in Newtonian Mechanics or General Relativity. These corrections generically violate the Equivalence Principle. The GUP has also been indirectly applied to the gravitational source by relating the GUP modified Hawking temperature to a deformation of the background metric. Such a deformed background metric determines new geodesic motions without violating the Equivalence Principle. We point out her…
The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We present tighter bounds on both entropic uncertainty relation and information exclusion principle for multiple measurements in the presence of quantum memory. As applications, three incompatible measurements on Werner state and Horodecki's bound entangled state are investigated in details.
This paper deduces universal uncertainty principle in different quantum theories after about one century of proposing uncertainty principle by Heisenberg, i.e., new universal uncertainty principle of any orders of physical quantities in quantum physics, overcomes the difficulty that current quantum computer, quantum communication, quantum control, quantum mechanics and so on theories cannot give exact values of general uncertainty of any orders of physical quantities, further gives all relevant different expressions of the universal uncertainty principle and their applications. In fact, our st…
It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieliński [J. Phys. A 35 (2002) 1405] is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (UR's) on a circle are that based on the Gram-Robertson matrix. Several of these generalized UR's are displayed and related criterions for squeezed states are discussed.
It is shown that the Bohm equations for the phase $S$ and squared modulus $ρ$ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum $p_s$ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat grad$S$ and $p_s$ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schroedinger type on the examples of the stochastic image …
We show that three principle means of treating privacy amplification in quantum key distribution, private state distillation, classical privacy amplification, and via the uncertainty principle, are equivalent and interchangeable. By adapting the security proof based on the uncertainty principle, we construct a new protocol for private state distillation which we prove is identical to standard classical privacy amplification. Underlying this approach is a new characterization of private states, related to their standard formulation by the uncertainty principle, which gives a more physical under…
We present an "uncertainty principle" for quantum channels, showing a relationship between the dimensions of the range of a channel and the range of its complement. We examine some interesting specific cases, and discuss consequences for privacy of quantum channels.
Application of the uncertainty principle to conditional measurements is investigated, and found to be valid for measurements on separated sub-systems. In light of this, an apparent violation of the uncertainty principle obtained by Kim and Shih in their realization of Popper's experiment (quant-ph/9905039) is explained through analogy with a simple optical system.
The uncertainty principle, which bounds the uncertainties involved in obtaining precise outcomes for two complementary variables defining a quantum particle, is a crucial aspect in quantum mechanics. Recently, the uncertainty principle in terms of entropy has been extended to the case involving quantum entanglement. With previously obtained quantum information for the particle of interest, the outcomes of both non-commuting observables can be predicted precisely, which greatly generalises the uncertainty relation. Here, we experimentally investigated the entanglement-assisted entropic uncertai…
The local uncertainty relation (LUR) criteria for quantum entanglement, which is dependent on chosen observables, is developed recent. In the paper, applying the uncertainty principle, an entanglement criteria for multipartite Gaussian states is given, which is implemented by a minimum optimization computer program and independent on observalbes.
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
The usual Heisenberg uncertainty relation for position and momentum may be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty. This "exact" uncertainty relation is valid for_all_ pure states, and is sufficiently strong to provide an axiomatic basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schroedinger equation.
Beginners studying quantum mechanics are often baffled with electron tunneling.Hence an easy approach for comprehension of the topic is presented here on the basis of uncertainty principle.An estimate of the tunneling time is also derived from the same method.
We discuss the relation between density matrices and the uncertainty principle; this allows us to justify and explain a recent statement by Man'ko et al. We thereafter use Hardy's uncertainty principle to prove a new result for Wigner distributions dominated by a Gaussian and we relate this result to the coarse-graining of phase-space by "quantum blobs".
An exact uncertainty principle, formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrodinger equation. Thus there is an exact formulation of the uncertainty principle which precisely captures the essence of what is "quantum" about quantum mechanics.
Shannon Entropy Quantum Entanglement Criterion and the Generalized Uncertainty Principle
本文应用广义不确定性原理修正香农熵纠缠判据，发现其上界增大，并探讨了相关基本概念，揭示了量子引力对纠缠条件的影响。
Starting with the first-order singular Lagrangian describing the dynamical system with 2nd-class constraints, the noncommutative quantum mechanics on a curved space is investigated by the constraint star-product quantization formalism of the projection operator method. Imposing the additional constraints to eliminate the reduntant degrees of freedom, it is shown that the resultant noncommutative quantum system on the curved space is represented with two kinds of the constrained quantum systems, which are equivalent with each other. Then, it is shown that the resultant Hamiltonians contain the …
Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics. The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space. However, many physical systems are dynamic in nature and described with the formalism of a quantum channel. In this paper, we show that the uncertainty principle can be reformulated to include p…
Historically, the element of uncertainty in quantum mechanics has been expressed through mathematical identities called uncertainty relations, a great many of which continue to be discovered. These relations use diverse measures to quantify uncertainty (and joint uncertainty). In this paper we use operational information-theoretic principles to identify the common essence of all such measures, thereby defining measure-independent notions of uncertainty and joint uncertainty. We find that most existing entropic uncertainty relations use measures of joint uncertainty that yield themselves to a s…
Some aspects of application of the Uncertainty Principle in the range of interaction radiation with matter surveyed. The procedure of adjustment is proposed at calculation of values of an electromagnetic energy in a quantum theory of a field.
We rederive uncertainty relations for the angular position and momentum of a particle on a circle by employing the exponential of the angle instead of the angle itself, which leads to circular variance as a natural measure of resolution. Intelligent states minimizing the uncertainty product under the constraint of a given uncertainty in angle or in angular momentum turn out to be given by Mathieu wave functions. We also discuss a number of physically feasible approximations to these optimal states. The theory is applied to the orbital angular momentum of a beam of photons and verified in an ex…
Supersymmetry in Quantum Mechanics by Generalized Uncertainty Principle
本文基于广义不确定性原理构造量子力学中的广义超对称性，利用微扰理论建立产生湮灭算符关系，并通过解析求解薛定谔方程生成新势。
In this note, we study the potential algebra for several models arising out of quantum mechanics with generalized uncertainty principle. We first show that the eigenvalue equation corresponding to the momentum-space Hamiltonian \[H=-(1+βp^{2})\frac{d}{dp}(1+βp^{2})\frac{d}{dp}+g(g-1)β^{2}p^{2}-gβ,\] which is associated with some one-dimensional models with minimal length uncertainty, can be solved by the unitary representations of the Lie algebra $\mathfrak{su}(2)$ if $g\in\{\tfrac{1}{2},1,\tfrac{3}{2},2,\cdots\}$. We then apply this result to spectral problems for the non-relativistic harmoni…
Uncertainty and Wigner negativity in Hilbert-space classical mechanics
本文在希尔伯特空间经典力学中重现了量子力学的不确定性关系和威格纳负值特征。
The aim of this paper is to derive a new uncertainty principle for the generalized $q$-Bessel wavelet transform studied earlier in \cite{Rezguietal}. In this paper, an uncertainty principle associated with wavelet transforms in the $q$-calculus framework has been established. A two-parameters extension of the classical Bessel operator is applied to generate a wavelet function which is exploited next to explore a wavelet uncertainty principle already in the $q$-calculus framework.
This paper derives a new directional uncertainty principle for quaternion valued functions subject to the quaternion Fourier transformation. This can be generalized to establish directional uncertainty principles in Clifford geometric algebras with quaternion subalgebras. We demonstrate this with the example of a directional spacetime algebra function uncertainty principle related to multivector wave packets.
Resolving the Quantum Measurement Problem through Leveraging the Uncertainty Principle
针对量子测量问题，本文利用不确定性原理，通过相空间量子力学中的Wigner Moyal方程调节观测窗口，证明足够的不确定性可确保宇宙适定，测量将叠加态坍缩为唯一解，统一了经典与量子力学等概念。
The celebrated Heisenberg Uncertainty Principle Δx Δp\ge \hbar/2 can allow measurement accuracies less than Δx or Δp. Classical analog of this is known as sub-Fourier sensitivity. We illustrate this phenomenon in a step by step process using the example of compass state, as suggested by Zurek.
The entropic uncertainty principle as outlined by Maassen and Uffink for a pair of non-degenerate observables in a finite level qusystem is generalized here to the case of a pair of arbitrary quantum measurements. In particular, our result includes not only the case of projectivmeasurements (or equivalently, observables) exhibiting degeneracy but also an uncertainty principle for a single measurement.
A decision-theoretic approach to dealing with uncertainty in quantum mechanics
本文提出决策理论框架以处理量子力学中的不确定性，通过将测量视为具有不确定结果的行动，将Born规则纳入效用函数，从而将精确概率理论与量子力学分离。
On Wigdersons' approach to the uncertainty principle
本文基于 Wigdersons 的视角重新审视不确定性原理，通过引入更一般的基本不确定性原理，推导出多种新不等式并证明了作者提出的高维推广猜想。
We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the supports of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state independent uncertainty relations. Our res…
Quantum uncertainty is described here in two guises: indeterminacy with its concomitant indeterminism of measurement outcomes, and fuzziness, or unsharpness. Both features were long seen as obstructions of experimental possibilities that were available in the realm of classical physics. The birth of quantum information science was due to the realization that such obstructions can be turned into powerful resources. Here we review how the utilization of quantum fuzziness makes room for a notion of approximate joint measurement of noncommuting observables. We also show how from a classical perspe…
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